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P systems simulating oracle computations Antonio E. Porreca Alberto Leporati Giancarlo Mauri Claudio Zandron Dipartimento di Informatica, Sistemistica e Comunicazione Università degli Studi di Milano-Bicocca, Italy 12th Conference on Membrane Computing Fontainebleau, France, 24 August 2011

P systems simulating oracle computations

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Page 1: P systems simulating oracle computations

P systems simulatingoracle computations

Antonio E. Porreca Alberto Leporati Giancarlo MauriClaudio Zandron

Dipartimento di Informatica, Sistemistica e ComunicazioneUniversità degli Studi di Milano-Bicocca, Italy

12th Conference on Membrane ComputingFontainebleau, France, 24 August 2011

Page 2: P systems simulating oracle computations

Summary

I We show how to reuse existing recogniser P systemsas “subroutines”

I This allows us to simulate oraclesI The procedure is quite general

(though technical details may vary)I As an application, we improve the lower bound

on PMCAM(−d,−n) to PPP

2/16

Page 3: P systems simulating oracle computations

P systems with active membranes

I Known for their ability to solvecomputationally hard problems

I Here we focus on restricted elementary membranes(no nonelementary division, no dissolution)

Object evolution [a→ w]αhCommunication (send-in) a [ ]αh → [b]βhCommunication (send-out) [a]αh → [ ]βh bElementary division [a]αh → [b]βh [c]γh

3/16

Page 4: P systems simulating oracle computations

Uniform families of recogniser P systemsI For each input length n = |x| we construct a P system Πn

receiving as input a multiset encoding xI Both are constructed by fixed polytime Turing machinesI The resulting P system decides if x ∈ L

M1

1 11

x ∈ Σ?M2

0 10

Y E S

N O

aab

aab

Input multiset

n ∈ N

4/16

Page 5: P systems simulating oracle computations

The complexity class PMCAM(−d,−n)

It consists of the languages recognised in polytime by uniformfamilies of P systems with restricted elementary membranes

I Contains NP problems [Zandron et al. 2000](semi-uniform solution)

I Contains NP problems [Pérez-Jiménez et al. 2003](first uniform solution)

I Is contained in PSPACE [Sosík, Rodríguez-Patón 2007]I Contains PP problems [Porreca et al. 2010, 2011]

On the other hand, by using nonelementary division(class PMCAM) we obtain exactly PSPACE

5/16

Page 6: P systems simulating oracle computations

Solving 3SATIs ϕ(x1, x2, x3) satisfiable?

x2x3x1

6/16

Page 7: P systems simulating oracle computations

Solving 3SATIs ϕ(x1, x2, x3) satisfiable?

x2x3x1

6/16

Page 8: P systems simulating oracle computations

Solving 3SATIs ϕ(x1, x2, x3) satisfiable?

t2x3x1 f2x3x1

6/16

Page 9: P systems simulating oracle computations

Solving 3SATIs ϕ(x1, x2, x3) satisfiable?

t2x3x1 f2x3x1

6/16

Page 10: P systems simulating oracle computations

Solving 3SATIs ϕ(x1, x2, x3) satisfiable?

t2x3f1 f2 t3x1t2x3t1 f2 f3x1

6/16

Page 11: P systems simulating oracle computations

Solving 3SATIs ϕ(x1, x2, x3) satisfiable?

t2x3f1 f2 t3x1t2x3t1 f2 f3x1

6/16

Page 12: P systems simulating oracle computations

Solving 3SATIs ϕ(x1, x2, x3) satisfiable?

t2 t3f1 f2 t3t1t2 t3t1 f2 f3t1

t2 f3f1 f2 t3f1t2 f3t1 f2 f3f1

6/16

Page 13: P systems simulating oracle computations

Solving 3SATIs ϕ(x1, x2, x3) satisfiable?

t2 t3f1 f2 t3t1t2 t3t1 f2 f3t1

t2 f3f1 f2 t3f1t2 f3t1 f2 f3f1

6/16

Page 14: P systems simulating oracle computations

Solving 3SATIs ϕ(x1, x2, x3) satisfiable?

t2 t3f1 f2 t3t1t2 t3t1 f2 f3t1

t2 f3f1 f2 t3f1t2 f3t1 f2 f3f1

t t t t

6/16

Page 15: P systems simulating oracle computations

Solving 3SATIs ϕ(x1, x2, x3) satisfiable?

t2 t3f1 f2 t3t1t2 t3t1 f2 f3t1

t2 f3f1 f2 t3f1t2 f3t1 f2 f3f1

t t t t

6/16

Page 16: P systems simulating oracle computations

Solving 3SATIs ϕ(x1, x2, x3) satisfiable?

t2 t3f1 f2 t3t1t2 t3t1 f2 f3t1

t2 f3f1 f2 t3f1t2 f3t1 f2 f3f1

t t tY E S

6/16

Page 17: P systems simulating oracle computations

The complexity class PP

DefinitionL ∈ PP if it is accepted in polytime by a nondeterministic TMsuch that more than half of its computations are accepting

Solving PP is “essentially the same as” countingthe number of solutions

Problem (T H R E S H O L D -3SAT)Given a Boolean formula ϕ over m variables and an integerk < 2m, do more than k assignments out of 2m satisfy ϕ?

TheoremT H R E S H O L D -3SAT is PP-complete

7/16

Page 18: P systems simulating oracle computations

Solving T H R E S H O L D -3SATDoes ϕ(x1, x2, x3) have more than 3 satisfying assignments?

t2 t3f1 f2 t3t1t2 t3t1 f2 f3t1

t2 f3f1 f2 t3f1t2 f3t1 f2 f3f1

8/16

Page 19: P systems simulating oracle computations

Solving T H R E S H O L D -3SATDoes ϕ(x1, x2, x3) have more than 3 satisfying assignments?

t2 t3f1 f2 t3t1t2 t3t1 f2 f3t1

t2 f3f1 f2 t3f1t2 f3t1 f2 f3f1

0

8/16

Page 20: P systems simulating oracle computations

Solving T H R E S H O L D -3SATDoes ϕ(x1, x2, x3) have more than 3 satisfying assignments?

t2 t3f1 f2 t3t1t2 t3t1 f2 f3t1

t2 f3f1 f2 t3f1t2 f3t1 f2 f3f1

0 0

8/16

Page 21: P systems simulating oracle computations

Solving T H R E S H O L D -3SATDoes ϕ(x1, x2, x3) have more than 3 satisfying assignments?

t2 t3f1 f2 t3t1t2 t3t1 f2 f3t1

t2 f3f1 f2 t3f1t2 f3t1 f2 f3f1

0 0 0

8/16

Page 22: P systems simulating oracle computations

Solving T H R E S H O L D -3SATDoes ϕ(x1, x2, x3) have more than 3 satisfying assignments?

t2 t3f1 f2 t3t1t2 t3t1 f2 f3t1

t2 f3f1 f2 t3f1t2 f3t1 f2 f3f1

t t t t0 0 0

8/16

Page 23: P systems simulating oracle computations

Solving T H R E S H O L D -3SATDoes ϕ(x1, x2, x3) have more than 3 satisfying assignments?

t2 t3f1 f2 t3t1t2 t3t1 f2 f3t1

t2 f3f1 f2 t3f1t2 f3t1 f2 f3f1

t t t t0 0 0

8/16

Page 24: P systems simulating oracle computations

Solving T H R E S H O L D -3SATDoes ϕ(x1, x2, x3) have more than 3 satisfying assignments?

t2 t3f1 f2 t3t1t2 t3t1 f2 f3t1

t2 f3f1 f2 t3f1t2 f3t1 f2 f3f1

t t t

t− − −

8/16

Page 25: P systems simulating oracle computations

Solving T H R E S H O L D -3SATDoes ϕ(x1, x2, x3) have more than 3 satisfying assignments?

t2 t3f1 f2 t3t1t2 t3t1 f2 f3t1

t2 f3f1 f2 t3f1t2 f3t1 f2 f3f1

t t t

t− − −

8/16

Page 26: P systems simulating oracle computations

Solving T H R E S H O L D -3SATDoes ϕ(x1, x2, x3) have more than 3 satisfying assignments?

t2 t3f1 f2 t3t1t2 t3t1 f2 f3t1

t2 f3f1 f2 t3f1t2 f3t1 f2 f3f1

t t t

Y E S− − −

8/16

Page 27: P systems simulating oracle computations

Simulating Turing machines I

q

0 10

S

0− + − 0 0

0 1 2 3 4

H2,q

9/16

Page 28: P systems simulating oracle computations

Simulating Turing machines II

δ(q1, 0) = (q2, 1, .)

{Hq1,i, [ ]−i → [Hq2,i+1]+i[Hq2,i+1]+i → [ ]+i Hq2,i+1

S

0− + − 0 0

0 1 2 3 4

H2,q1

10/16

Page 29: P systems simulating oracle computations

Simulating Turing machines II

δ(q1, 0) = (q2, 1, .)

{Hq1,i, [ ]−i → [Hq2,i+1]+i[Hq2,i+1]+i → [ ]+i Hq2,i+1

S

0− + − 0 0

0 1 2 3 4

H2,q1

10/16

Page 30: P systems simulating oracle computations

Simulating Turing machines II

δ(q1, 0) = (q2, 1, .)

{Hq1,i, [ ]−i → [Hq2,i+1]+i[Hq2,i+1]+i → [ ]+i Hq2,i+1

S

0− + + 0 0

0 1 2 3 4

H3,q2

10/16

Page 31: P systems simulating oracle computations

Simulating Turing machines II

δ(q1, 0) = (q2, 1, .)

{Hq1,i, [ ]−i → [Hq2,i+1]+i[Hq2,i+1]+i → [ ]+i Hq2,i+1

S

0− + + 0 0

0 1 2 3 4

H3,q2

10/16

Page 32: P systems simulating oracle computations

Simulating Turing machines II

δ(q1, 0) = (q2, 1, .)

{Hq1,i, [ ]−i → [Hq2,i+1]+i[Hq2,i+1]+i → [ ]+i Hq2,i+1

S

0− + + 0 0

0 1 2 3 4

H3,q2

10/16

Page 33: P systems simulating oracle computations

Using P systems as subroutines

S

0

− + − + 0

0 1 2 3 4

H2,q?

11/16

Page 34: P systems simulating oracle computations

Using P systems as subroutines

S

0

− + − + 0

0 1 2 3 4

H2,q?

Π Π Π Π Π

A

0

0 0 0 0 0

11/16

Page 35: P systems simulating oracle computations

Using P systems as subroutines

S

0

− + − + 0

0 1 2 3 4

H2,q?

Π Π Π Π Π

A

0

0 0 + 0 0

11/16

Page 36: P systems simulating oracle computations

Using P systems as subroutines

S

0

− + − + 0

0 1 2 3 4

H2,q?

Π Π Π Π Π

A

0

0 0 + 0 0

11/16

Page 37: P systems simulating oracle computations

Using P systems as subroutines

S

0

− + − + 0

0 1 2 3 4

H2,q?

Π Π Π Π Π

A

0

0 0 + 0 0

11/16

Page 38: P systems simulating oracle computations

Using P systems as subroutines

S

0

− + − + 0

0 1 2 3 4

H2,q?

Π Π Π Π Π

A

0

0 0 + 0 0

11/16

Page 39: P systems simulating oracle computations

Using P systems as subroutines

S

0

− + − + 0

0 1 2 3 4

H2,q?

Π Π Π Π Π

A

0

0 0 + 0 0

11/16

Page 40: P systems simulating oracle computations

Using P systems as subroutines

S

0

− + − + 0

0 1 2 3 4

H2,q?

Π Π Π Π Π

A

0

0 0 + 0 0

11/16

Page 41: P systems simulating oracle computations

Using P systems as subroutines

S

0

− + − + 0

0 1 2 3 4

H2,q?

Π Π Π Π Π

A

0

0 0 − 0 0

AY E S

11/16

Page 42: P systems simulating oracle computations

Using P systems as subroutines

S

0

− + − + 0

0 1 2 3 4

H2,q?

Π Π Π Π Π

A

0

0 0 − 0 0

AY E S

11/16

Page 43: P systems simulating oracle computations

Using P systems as subroutines

S

0

− + − + 0

0 1 2 3 4

H2,q?

Π Π Π Π Π

A

+

0 0 − 0 0

11/16

Page 44: P systems simulating oracle computations

Using P systems as subroutines

S

0

− + − + 0

0 1 2 3 4

H2,q?

Π Π Π Π Π

A

+

0 0 − 0 0

11/16

Page 45: P systems simulating oracle computations

Main result

TheoremPPP ⊆ PMCAM(−d,−n)

Proof.I Any polytime TM M with a PP oracle can be simulated

by a polytime TM M′ with an oracle forT H R E S H O L D -3SAT and only one tape

I Just apply a reduction (which always exists,since T H R E S H O L D -3SAT is PP-complete)before querying the oracle

I And we know how to simulate the TM M′

with a polytime AM(−d,−n) uniform family.

12/16

Page 46: P systems simulating oracle computations

Discussion I

I We can solve QSAT (PSPACE-complete)by using nonelementary divisionand a membrane structure of depth Θ(n)

I QSAT instances have an arbitrary numberof alternations of quantifiers

I By fixing the first quantifier (∀ or ∃) and the numberof alternations, we get complete problems for all levelsof the polynomial hierarchy

I Formulae with k alternations can be solved by P systemsusing nonelementary division and a membrane structureof depth Θ(k)

I Notice that k does not depend on the input size

13/16

Page 47: P systems simulating oracle computations

Discussion II

I Let PH be the union of the levelsof the polynomial hierarchy

I Toda’s theorem tells us that PH ⊆ PPP

I So we also have PH ⊆ PMCAM(−d,−n)

I This means that all levels of the polynomial hierarchycan be solved by using P systems with only elementarydivision and membrane structure of depth 3

I Does this mean PSPACE ⊆ PMCAM(−d,−n)

and so PSPACE = PMCAM(−d,−n)?I Not immediately: PH is not known neither conjectured

to be PSPACE

14/16

Page 48: P systems simulating oracle computations

Open problems

I Prove that we can always do the oracle simulationI If we can reset the “oracle P systems”

then we only need a single copy of itI It might still be possible that PMCAM(−d,−n) = PSPACE

even if PH 6= PSPACEI But maybe it would be more interesting if it turns out

that PMCAM(−d,−n) = PPP

15/16

Page 49: P systems simulating oracle computations

Merci de votre attention!

Thanks for your attention!